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## G = Dic3×C4○D4order 192 = 26·3

### Direct product of Dic3 and C4○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Dic3×C4○D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — Dic3×C4○D4
 Lower central C3 — C6 — Dic3×C4○D4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for Dic3×C4○D4
G = < a,b,c,d,e | a6=c4=e2=1, b2=a3, d2=c2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 552 in 310 conjugacy classes, 195 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C4×C4○D4, C2×C4×Dic3, C23.26D6, D4×Dic3, Q8×Dic3, C6×C4○D4, Dic3×C4○D4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C24, C2×Dic3, C22×S3, C23×C4, C2×C4○D4, C22×Dic3, S3×C23, C4×C4○D4, S3×C4○D4, C23×Dic3, Dic3×C4○D4

Smallest permutation representation of Dic3×C4○D4
On 96 points
Generators in S96
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 79 4 82)(2 84 5 81)(3 83 6 80)(7 20 10 23)(8 19 11 22)(9 24 12 21)(13 85 16 88)(14 90 17 87)(15 89 18 86)(25 91 28 94)(26 96 29 93)(27 95 30 92)(31 56 34 59)(32 55 35 58)(33 60 36 57)(37 65 40 62)(38 64 41 61)(39 63 42 66)(43 68 46 71)(44 67 47 70)(45 72 48 69)(49 76 52 73)(50 75 53 78)(51 74 54 77)
(1 40 15 35)(2 41 16 36)(3 42 17 31)(4 37 18 32)(5 38 13 33)(6 39 14 34)(7 67 94 77)(8 68 95 78)(9 69 96 73)(10 70 91 74)(11 71 92 75)(12 72 93 76)(19 46 30 50)(20 47 25 51)(21 48 26 52)(22 43 27 53)(23 44 28 54)(24 45 29 49)(55 82 65 86)(56 83 66 87)(57 84 61 88)(58 79 62 89)(59 80 63 90)(60 81 64 85)
(1 32 15 37)(2 33 16 38)(3 34 17 39)(4 35 18 40)(5 36 13 41)(6 31 14 42)(7 70 94 74)(8 71 95 75)(9 72 96 76)(10 67 91 77)(11 68 92 78)(12 69 93 73)(19 43 30 53)(20 44 25 54)(21 45 26 49)(22 46 27 50)(23 47 28 51)(24 48 29 52)(55 89 65 79)(56 90 66 80)(57 85 61 81)(58 86 62 82)(59 87 63 83)(60 88 64 84)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 86)(8 87)(9 88)(10 89)(11 90)(12 85)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 54)(38 49)(39 50)(40 51)(41 52)(42 53)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)

G:=sub<Sym(96)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,79,4,82)(2,84,5,81)(3,83,6,80)(7,20,10,23)(8,19,11,22)(9,24,12,21)(13,85,16,88)(14,90,17,87)(15,89,18,86)(25,91,28,94)(26,96,29,93)(27,95,30,92)(31,56,34,59)(32,55,35,58)(33,60,36,57)(37,65,40,62)(38,64,41,61)(39,63,42,66)(43,68,46,71)(44,67,47,70)(45,72,48,69)(49,76,52,73)(50,75,53,78)(51,74,54,77), (1,40,15,35)(2,41,16,36)(3,42,17,31)(4,37,18,32)(5,38,13,33)(6,39,14,34)(7,67,94,77)(8,68,95,78)(9,69,96,73)(10,70,91,74)(11,71,92,75)(12,72,93,76)(19,46,30,50)(20,47,25,51)(21,48,26,52)(22,43,27,53)(23,44,28,54)(24,45,29,49)(55,82,65,86)(56,83,66,87)(57,84,61,88)(58,79,62,89)(59,80,63,90)(60,81,64,85), (1,32,15,37)(2,33,16,38)(3,34,17,39)(4,35,18,40)(5,36,13,41)(6,31,14,42)(7,70,94,74)(8,71,95,75)(9,72,96,76)(10,67,91,77)(11,68,92,78)(12,69,93,73)(19,43,30,53)(20,44,25,54)(21,45,26,49)(22,46,27,50)(23,47,28,51)(24,48,29,52)(55,89,65,79)(56,90,66,80)(57,85,61,81)(58,86,62,82)(59,87,63,83)(60,88,64,84), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,86)(8,87)(9,88)(10,89)(11,90)(12,85)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,54)(38,49)(39,50)(40,51)(41,52)(42,53)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,79,4,82)(2,84,5,81)(3,83,6,80)(7,20,10,23)(8,19,11,22)(9,24,12,21)(13,85,16,88)(14,90,17,87)(15,89,18,86)(25,91,28,94)(26,96,29,93)(27,95,30,92)(31,56,34,59)(32,55,35,58)(33,60,36,57)(37,65,40,62)(38,64,41,61)(39,63,42,66)(43,68,46,71)(44,67,47,70)(45,72,48,69)(49,76,52,73)(50,75,53,78)(51,74,54,77), (1,40,15,35)(2,41,16,36)(3,42,17,31)(4,37,18,32)(5,38,13,33)(6,39,14,34)(7,67,94,77)(8,68,95,78)(9,69,96,73)(10,70,91,74)(11,71,92,75)(12,72,93,76)(19,46,30,50)(20,47,25,51)(21,48,26,52)(22,43,27,53)(23,44,28,54)(24,45,29,49)(55,82,65,86)(56,83,66,87)(57,84,61,88)(58,79,62,89)(59,80,63,90)(60,81,64,85), (1,32,15,37)(2,33,16,38)(3,34,17,39)(4,35,18,40)(5,36,13,41)(6,31,14,42)(7,70,94,74)(8,71,95,75)(9,72,96,76)(10,67,91,77)(11,68,92,78)(12,69,93,73)(19,43,30,53)(20,44,25,54)(21,45,26,49)(22,46,27,50)(23,47,28,51)(24,48,29,52)(55,89,65,79)(56,90,66,80)(57,85,61,81)(58,86,62,82)(59,87,63,83)(60,88,64,84), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,86)(8,87)(9,88)(10,89)(11,90)(12,85)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,54)(38,49)(39,50)(40,51)(41,52)(42,53)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96) );

G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,79,4,82),(2,84,5,81),(3,83,6,80),(7,20,10,23),(8,19,11,22),(9,24,12,21),(13,85,16,88),(14,90,17,87),(15,89,18,86),(25,91,28,94),(26,96,29,93),(27,95,30,92),(31,56,34,59),(32,55,35,58),(33,60,36,57),(37,65,40,62),(38,64,41,61),(39,63,42,66),(43,68,46,71),(44,67,47,70),(45,72,48,69),(49,76,52,73),(50,75,53,78),(51,74,54,77)], [(1,40,15,35),(2,41,16,36),(3,42,17,31),(4,37,18,32),(5,38,13,33),(6,39,14,34),(7,67,94,77),(8,68,95,78),(9,69,96,73),(10,70,91,74),(11,71,92,75),(12,72,93,76),(19,46,30,50),(20,47,25,51),(21,48,26,52),(22,43,27,53),(23,44,28,54),(24,45,29,49),(55,82,65,86),(56,83,66,87),(57,84,61,88),(58,79,62,89),(59,80,63,90),(60,81,64,85)], [(1,32,15,37),(2,33,16,38),(3,34,17,39),(4,35,18,40),(5,36,13,41),(6,31,14,42),(7,70,94,74),(8,71,95,75),(9,72,96,76),(10,67,91,77),(11,68,92,78),(12,69,93,73),(19,43,30,53),(20,44,25,54),(21,45,26,49),(22,46,27,50),(23,47,28,51),(24,48,29,52),(55,89,65,79),(56,90,66,80),(57,85,61,81),(58,86,62,82),(59,87,63,83),(60,88,64,84)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,86),(8,87),(9,88),(10,89),(11,90),(12,85),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,54),(38,49),(39,50),(40,51),(41,52),(42,53),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)]])

60 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 4S ··· 4AD 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 ··· 2 3 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 ··· 2 2 1 1 1 1 2 ··· 2 3 ··· 3 6 ··· 6 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 Dic3 C4○D4 S3×C4○D4 kernel Dic3×C4○D4 C2×C4×Dic3 C23.26D6 D4×Dic3 Q8×Dic3 C6×C4○D4 C3×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 Dic3 C2 # reps 1 3 3 6 2 1 16 1 3 3 1 8 8 4

Matrix representation of Dic3×C4○D4 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 3 0 0 0 0 3 9
,
 8 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 12 11 0 0 0 0 1
,
 1 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 5 8 0 0 0 0 8 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 11 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,3,3,0,0,0,0,9],[8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,11,1],[1,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,5,0,0,0,0,8,8,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,11,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

Dic3×C4○D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_4\circ D_4
% in TeX

G:=Group("Dic3xC4oD4");
// GroupNames label

G:=SmallGroup(192,1385);
// by ID

G=gap.SmallGroup(192,1385);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,570,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=c^4=e^2=1,b^2=a^3,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations

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